TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Extension-theoretic interpretation of the standard five-term exact sequence

Statement

For an extension $1 \to N \to G \to Q \to 1$ and an abelian $G$ -module $A$ , assume the standard low-degree sequence

0 \to H^{1} (Q, A^{N}) \overset{Inf}{\to} H^{1} (G, A) \overset{Res}{\to} H^{1} (N, A)^{Q} \overset{Tra}{\to} H^{2} (Q, A^{N}) \overset{Inf}{\to} H^{2} (G, A)

is exact. Then the transgression detects extension of degree-one classes: for $[u] \in H^{1} (N, A)^{Q}$ one has $Tra [u] = 0$ exactly when $[u]$ is the restriction of a class in $H^{1} (G, A)$ . Under the factor-set classification, the last inflation map is represented by pulling a $Q$ -extension back along $G \to Q$ and then pushing it out along the inclusion $A^{N} ↪ A$ .

Facts & Assumptions

Given: An extension $1 \to N \to G \to Q \to 1$ and an abelian $G$ -module $A$ .

[F1]

Restriction and inflation in degree one are the explicit maps defined on cocycles in Restriction, inflation, and the quotient conjugation action on first cohomology.

[L1]

The degree-one inflation-restriction sequence is exact (Inflation-restriction exact sequence in degree one).

[L2]

$H^{2} (Q, A^{N})$ classifies extensions of $Q$ by $A^{N}$ (H^2 classifies extensions with fixed abelian kernel action).

[A1]

The displayed five-term sequence is the standard exact low-degree five-term sequence attached to $1 \to N \to G \to Q \to 1$ .

Proof

technique · direct

1.1

The first three terms are exactly the degree-one inflation-restriction sequence, so they are exact by [L1], with maps described concretely by [F1].

F1L1given

1.2

Under [L2], a class in $H^{2} (Q, A^{N})$ is an extension of $Q$ by $A^{N}$ . The map to $H^{2} (G, A)$ first pulls that extension back along $G \to Q$ , producing an extension of $G$ by $A^{N}$ , and then pushes out along the inclusion $A^{N} ↪ A$ . That is the extension-theoretic meaning of the last inflation map in the displayed sequence.

L2givenalgebra

1.3

Exactness of [A1] at $H^{1} (N, A)^{Q}$ says $\ker (Tra) = im (Res) .$ Thus $Tra [u] = 0$ exactly when $[u]$ is the restriction of a degree-one class on $G$ , which is precisely the asserted extension criterion for the cohomology class.

A1algebra

2.1

Steps 1.2 and 1.3 give the two claimed interpretations, while step 1.1 identifies the preceding degree-one maps.

step 1.1step 1.2step 1.3∎

Depends on

Used by

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Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)